Optimality Conditions In Terms Research Articles

A note on optimality conditions in terms of convexificators for a bilevel multiobjective optimization problem

Necessary optimality conditions for a vectorial bilevel programming problem are given by Dempe et al. [Optimization 69, 1811–1830, 2020]. The main result, Theorem 1, is based on a result of Dutta and Chandra [J. Optim. Theory Appl. 113, 41–64, 2002] which turned out to be incorrect. In this work, we propose an improvement to Dempe et al.'s result that does not use any convexity-type restriction.

Field- and concentration-dependent relaxation of magnetic nanoparticles and optimality conditions for magnetic fluid hyperthermia

The field-dependent relaxation dynamics of suspended magnetic nanoparticles continues to present a fascinating topic of basic science that at the same time is highly relevant for several technological and biomedical applications. Renewed interest in the intriguing behavior of magnetic nanoparticles in response to external fields has at least in parts be driven by rapid advances in magnetic fluid hyperthermia research. Although a wealth of experimental, theoretical, and simulation studies have been performed in this field in recent years, several contradictory findings have so far prevented the emergence of a consistent picture. Here, we present a dynamic mean-field theory together with comprehensive computer simulations of a microscopic model system to systematically discuss the influence of several key parameters on the relaxation dynamics, such as steric and dipolar interactions, the external magnetic field strength and frequency, as well as the ratio of Brownian and Néel relaxation time. We also discuss the specific and intrinsic loss power as measures of the efficiency of magnetic fluid heating and discuss optimality conditions in terms of fluid and field parameters. Our results are helpful to reconcile contradictory findings in the literature and provide an important step towards a more consistent understanding. In addition, our findings also help to select experimental conditions that optimize magnetic fluid heating applications.

Optimal control of a reaction–diffusion model related to the spread of COVID-19

This paper is concerned with the well-posedness and optimal control problem of a reaction–diffusion system for an epidemic susceptible–exposed–infected–recovered–susceptible mathematical model in which the dynamics develops in a spatially heterogeneous environment. Using as control variables the transmission rates [Formula: see text] and [Formula: see text] of contagion resulting from the contact with both asymptomatic and symptomatic persons, respectively, we optimize the number of exposed and infected individuals at a final time [Formula: see text] of the controlled evolution of the system. More precisely, we search for the optimal [Formula: see text] and [Formula: see text] such that the number of infected plus exposed does not exceed at the final time a threshold value [Formula: see text], fixed a priori. We prove here the existence of optimal controls in a proper functional framework and we derive the first-order necessary optimality conditions in terms of the adjoint variables.

Erratum to: optimality conditions in terms of convexificators for a bilevel multiobjective optimization problem

Erratum to: optimality conditions in terms of convexificators for a bilevel multiobjective optimization problem

Applying tangential subdifferentials in bilevel optimization

The problem considered in this paper is a sequence of two optimization problems in which the feasible region of the upper-level optimization problem is determined implicitly by the solution set of the lower-level optimization problem. Using the optimal value reformulation, together with the partial calmness property, we give necessary optimality conditions in terms of tangential subdifferentials. An example that illustrates our finding is also given.

Directional Differentiability, Coexhausters, Codifferentials and Polyhedral DC Functions

Codifferentials and coexhausters are used to describe nonhomogeneous approximations of a nonsmooth function. Despite the fact that coexhausters are modern generalizations of codifferentials, the theories of these two concepts continue to develop simultaneously. Moreover, codifferentials and coexhausters are strongly connected with DC functions. In this paper we trace analogies between all these objects, and prove the equivalence of the boundedness and optimality conditions described in terms of these notions. This allows one to extend the results derived in terms of one object to the problems stated via the other one. Another contribution of this paper is the study of connection between nonhomogeneous approximations and directional derivatives and formulate optimality conditions in terms of nonhomogeneous approximations.

Optimality conditions and duality results for a robust bi-level programming problem

Robust bi-level programming problems are a newborn branch of optimization theory. In this study, we have considered a bi-level model with constraint-wise uncertainty at the upper-level, and the lower-level problem is fully convex. We use the optimal value reformulation to transform the given bi-level problem into a single-level mathematical problem and the concept of robust counterpart optimization to deal with uncertainty in the upper-level problem. Necessary optimality conditions are beneficial because any local minimum must satisfy these conditions. As a result, one can only look for local (or global) minima among points that hold the necessary optimality conditions. Here we have introduced an extended non-smooth robust constraint qualification (RCQ) and developed the KKT type necessary optimality conditions in terms of convexifactors and subdifferentials for the considered uncertain two-level problem. Further, we establish as an application the robust bi-level Mond-Weir dual (MWD) for the considered problem and produce the duality results. Moreover, an example is proposed to show the applicability of necessary optimality conditions.

Optimality Conditions for Mathematical Programs with Orthogonality Type Constraints

We consider the class of mathematical programs with orthogonality type constraints. Orthogonality type constraints appear by reformulating the sparsity constraint via auxiliary binary variables and relaxing them afterwards. For mathematical programs with orthogonality type constraints a necessary optimality condition in terms of T-stationarity is stated. The justification of T-stationarity is threefold. First, it allows to capture the global structure of mathematical programs with orthogonality type constraints in terms of Morse theory, i. e. deformation and cell-attachment results are established. For that, nondegeneracy for the T-stationary points is introduced and shown to hold generically. Second, we prove that Karush-Kuhn-Tucker points of the Scholtes-type regularization converge to T-stationary points of mathematical programs with orthogonality type constraints. This is done under the tailored linear independence constraint qualification, which turns out to be a generic property too. Third, we show that T-stationarity applied to the relaxation of sparsity constrained nonlinear optimization naturally leads to its M-stationary points. Moreover, we argue that all T-stationary points of this relaxation become degenerate.

Optimal control of a nonconserved phase field model of Caginalp type with thermal memory and double obstacle potential

In this paper, we investigate optimal control problems for a nonlinear state system which constitutes a version of the Caginalp phase field system modeling nonisothermal phase transitions with a nonconserved order parameter that takes thermal memory into account. The state system, which is a first-order approximation of a thermodynamically consistent system, is inspired by the theories developed by Green and Naghdi. It consists of two nonlinearly coupled partial differential equations that govern the phase dynamics and the universal balance law for internal energy, written in terms of the phase variable and the so-called thermal displacement, i.e., a primitive with respect to time of temperature. We extend recent results obtained for optimal control problems in which the free energy governing the phase transition was differentiable (i.e., of regular or logarithmic type) to the nonsmooth case of a double obstacle potential. As is well known, in this nondifferentiable case standard methods to establish the existence of appropriate Lagrange multipliers fail. This difficulty is overcome utilizing of the so-called deep quench approach. Namely, the double obstacle potential is approximated by a family of (differentiable) logarithmic ones for which the existence of optimal controls and first-order necessary conditions of optimality in terms of the adjoint state variables and a variational inequality are known. By proving appropriate bounds for the adjoint states of the approximating systems, we can pass to the limit in the corresponding first-order necessary conditions, thereby establishing meaningful first-order necessary optimality conditions also for the case of the double obstacle potential.

On multiobjective bilevel optimization using tangential subdifferentials

<p style='text-indent:20px;'>Multiobjective optimization problems typically have conflicting objectives, and a gain in one objective very often is an expense in another. In this paper, we investigate a multiobjective bilevel optimization problem. Using the concept of efficiency together with the optimal value reformulation, we give necessary optimality conditions in terms of tangential subdifferentials. Completely detailed first-order necessary optimality conditions are then derived in the special case where the lower level problem is convex. An example that illustrates our findings is also given.</p>

Second-Order Optimality Conditions for Strict Pareto Minima and Weak Efficiency for Nonsmooth Constrained Vector Equilibrium Problems

In this article, some types of lower and upper second-order strictly pseudoconvexity are provided for establishing sufficient conditions for the second-order strict local Pareto minima of nonsmooth vector equilibrium problem with set, inequality and equality constraints. Based on the notion of Gerstewitz mappings, some Kuhn-Tucker-type multiplier rules for the strict local Pareto minima of such problem are obtained. We also construct the second-order constraint qualification in terms of first- and second-order directional derivatives of the (CQ) and (CQ1) types. Using this constraint qualifications, some second-order primal and dual necessary optimality conditions in terms of second-order upper and lower Dini directional derivatives for such minima are derived. Under suitable assumptions on the lower and upper strictly pseudoconvexity of order two of objective and constraint functions, second-order necessary optimality conditions become sufficient optimality conditions to such problem. Some illustrative examples are also given for our findings.

Sufficient Optimality Conditions for a Robust Multiobjective Problem

In this note, we establish sufficient optimality conditions in terms of upper convexificators for a robust multiobjective optimization problem (UCP) involving nonsmooth/nonconvex real-valued functions. For this purpose, two kinds of generalized convex functions are defined in terms of upper convexificators.

Approximate solutions for robust multiobjective optimization programming in Asplund spaces

In this paper, we study a nonsmooth/nonconvex multiobjective optimization problem with uncertain constraints in arbitrary Asplund spaces. We first provide necessary optimality condition in a fuzzy form for approximate weakly robust efficient solutions and then establish necessary optimality theorem for approximate weakly robust quasi-efficient solutions of the problem in the sense of the limiting subdifferential by exploiting a fuzzy optimality condition in terms of the Fréchet subdifferential. Sufficient conditions for approximate (weakly) robust quasi-efficient solutions to such a problem are also driven under the new concept of generalized pseudo convex functions. Finally, we address an approximate Mond-Weir-type dual robust problem to the reference problem and explore weak, strong, and converse duality properties under assumptions of pseudo convexity.

Optimal H2 Moment Matching-Based Model Reduction for Linear Systems through (Non)convex Optimization

In this paper, we compute a (local) optimal reduced order model that matches a prescribed set of moments of a stable linear time-invariant system of high dimension. We fix the interpolation points and parametrize the models achieving moment-matching in a set of free parameters. Based on the parametrization and using the H2-norm of the approximation error as the objective function, we derive a nonconvex optimization problem, i.e., we search for the optimal free parameters to determine the model yielding the minimal H2-norm of the approximation error. Furthermore, we provide the necessary first-order optimality conditions in terms of the controllability and the observability Gramians of a minimal realization of the error system. We then propose two gradient-type algorithms to compute the (local) optimal models, with mathematical guarantees on the convergence. We also derive convex semidefinite programming relaxations for the nonconvex Problem, under the assumption that the error system admits block-diagonal Gramians, and derive sufficient conditions to guarantee the block diagonalization. The solutions resulting at each step of the proposed algorithms guarantee the achievement of the imposed moment matching conditions. The second gradient-based algorithm exhibits the additional property that, when stopped, yields a stable approximation with a reduced H2-error norm. We illustrate the theory on a CD-player and on a discretized heat equation.

Robust optimality conditions and duality for nonsmooth multiobjective fractional semi-infinite programming problems with uncertain data

In this article, some Karush-Kuhn-Tucker type robust optimality conditions and duality for an uncertain nonsmooth multiobjective fractional semi-infinite programming problem ((UMFP), for short) are established. First, we provide, by combining robust optimization and the robust limiting constraint qualification, robust necessary optimality conditions in terms of Mordukhovich's subdifferentials. Under suitable assumptions on the generalized convexity/the strictly generalized convexity, robust necessary optimality condition becomes robust sufficient optimality condition. Second, we formulate types of Mond-Weir and Wolfe robust dual problem for (UMFP) via the Mordukhovich subdifferentials. Finally, as an application, we establish weak/strong/converse robust duality theorems for the problem (UMFP) and its Mond-Weir and Wolfe types dual problem. Some illustrative examples are also provided for our findings.

Optimality conditions, approximate stationarity, and applications – a story beyond lipschitzness

Approximate necessary optimality conditions in terms of Frechet subgradients and normals for a rather general optimization problem with a potentially non-Lipschitzian objective function are established with the aid of Ekeland’s variational principle, the fuzzy Frechet subdifferential sum rule, and a novel notion of lower semicontinuity relative to a set-valued mapping or set. Feasible points satisfying these optimality conditions are referred to as approximately stationary. As applications, we derive a new general version of the extremal principle. Furthermore, we study approximate stationarity conditions for an optimization problem with a composite objective function and geometric constraints, a qualification condition guaranteeing that approximately stationary points of such a problem are M-stationary, and a multiplier-penalty-method which naturally computes approximately stationary points of the underlying problem. Finally, necessary optimality conditions for an optimal control problem with a non-Lipschitzian sparsity-promoting term in the objective function are established.

On the no-gap second-order optimality conditions for a non-smooth semilinear elliptic optimal control

This work is concerned with second-order necessary and sufficient optimality conditions for optimal control of a non-smooth semilinear elliptic partial differential equation, where the nonlinearity is the non-smooth max-function and thus the associated control-to-state operator is, in general, not Gâteaux-differentiable. In addition to standing assumptions, two main hypotheses are imposed. The first one is the Gâteaux-differentiability at the considered control of the objective functional and it is precisely characterized by the vanishing of an adjoint state on the set of all zeros of the corresponding state. The second one is a structural assumption on the sets of all points at which the values of the interested state are ‘close’ to the non-differentiability point of the max-function. We then derive a ‘no-gap’ theory of second-order optimality conditions in terms of a second-order generalized derivative of the cost functional, i.e. for which the only change between necessary and sufficient second-order optimality conditions are between a strict and non-strict inequality.

Optimality conditions in terms of contingent epiderivatives for strict local Pareto minima in vector optimization problems with constraints

Optimality conditions in terms of contingent epiderivatives for strict local Pareto minima in vector optimization problems with constraints

Neumann boundary optimal control problems governed by parabolic variational equalities

Abstract We consider a heat conduction problem S with mixed boundary conditions in an n-dimensional domain Ω with regular boundary and a family of problems Sα with also mixed boundary conditions in Ω, where α > 0 is the heat transfer coefficient on the portion of the boundary Γ1. In relation to these state systems, we formulate Neumann boundary optimal control problems on the heat flux q which is definite on the complementary portion Γ2 of the boundary of Ω. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system state and the adjoint state when the heat transfer coefficient α goes to infinity. Furthermore, we formulate particular boundary optimal control problems on a real parameter λ, in relation to the parabolic problems S and Sα and to mixed elliptic problems P and Pα . We find an explicit form for the optimal controls, we prove monotony properties and we obtain convergence results when the parameter time goes to infinity.

Metric Inequality Conditions on Sets and Consequences in Optimization

We study the implications of a well-known metric inequality condition on sets to the applicability of standard necessary optimality conditions for constrained optimization problems when a new constraint is added. We compare this condition with several other constraint qualification conditions in the literature, and due to its metric nature, we apply it to nonsmooth optimization problems in order to perform first a penalization and then to give optimality conditions in terms of generalized differentiability.